The convolution equation of Choquet and Deny on [IN]-groups

Harmonic functions on hypergroups

“…Throughout, C ru (G) denotes the Banach space of all bounded right uniformly continuous complex functions on G, where a bounded continuous function f on G is called right uniformly continuous if the map R f : y ∈ G → f * δ y ∈ C b (G) is continuous where C b (G) denotes the Banach space of bounded complex continuous functions on G. To avoid confusion, we remark that such a function on a group is called left uniformly continuous in [8,9,11].…”

Spectrum of a homogeneous graph

“…This condition imposes some structure on the group G acting on V . It turns out that a connected Cayley graph (G, K ) is invariant if and only if G is an [IN 0 ]-group as defined in [4]. A locally compact group G is called an [IN 0 ]-group if G = ∞ n=1 C n for some compact neighborhood C of the identity satisfying gC = C g for each g ∈ G. We first show the relationship between graph invariance and group structures.…”

Harmonic functions on topological groups and symmetric spaces

“…The essence of the condition is that, as in the finite dimensional case, the support of σ spreads out. The concept in the following definition is an extension of the one given for locally compact groups in [10] where a Liouville theorem is proved for [IN]-groups under the assumption of this notion. Definition 4.6 A tight measure σ on a metric group G is said to be non-singular with its translates if for each bounded set S in G, we have…”